For PSX, mine was probably Gran Turismo (1)...my roommate and I would stay up God-awful late playing this game just to get the next license, beat that track, tweak that next car (the Dodge concept car was tha bomb...), etc...
GT2 was similar, but just nothing compared to the original, in terms of just pure time spent playing it...it was better than the original, but the original was just so much better than any other racing game ever made, so it blew my mind...
On PS2, I fear that NBA Street may be the next big addiction...but otherwise it was Madden & SSX.